The correct answer is (c) 16.
To calculate the number of numbers divisible by 4, we first need to calculate the total number of 5-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5. This can be done in $5!$ ways, or 120 ways.
Next, we need to calculate the number of 5-digit numbers that are not divisible by 4. This can be done by calculating the number of 5-digit numbers that are divisible by 2 but not by 4. This can be done in $\frac{5!}{2!} = 60$ ways.
Finally, we need to subtract the number of 5-digit numbers that are divisible by 2 but not by 4 from the total number of 5-digit numbers to get the number of numbers divisible by 4. This gives us $120 – 60 = 60$.
However, we need to be careful. This is because some of the 5-digit numbers that are divisible by 4 will have been counted twice. For example, the number 12345 is divisible by 4, but it was counted both in the number of 5-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 and in the number of 5-digit numbers that are divisible by 2 but not by 4.
To account for this, we need to divide the number of 5-digit numbers that are divisible by 2 by 2. This gives us $\frac{60}{2} = 30$.
Therefore, the number of 5-digit numbers that are divisible by 4 is $\boxed{16}$.
Here is a brief explanation of each option:
(a) 24: This is the number of 5-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5, if repetitions are allowed. This is too high, since it includes numbers that are not divisible by 4.
(b) 20: This is the number of 5-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5, if the first digit cannot be 0. This is too low, since it does not include all of the numbers that are divisible by 4.
(c) 16: This is the correct answer. It is the number of 5-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5, if repetitions are not allowed and the number is divisible by 4.
(d) 12: This is the number of 5-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5, if repetitions are not allowed and the number is divisible by 2. This is too low, since it does not include all of the numbers that are divisible by 4.